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2 Growth: Linear vs ExponentialImagine two communities, Straightown and Powertown, each with an initial population of 10,000 people. Straightown grows at a constant rate of 500 people per year. Powertown grows at a constant rate of 5% per year.Compare the population growth of Straightown and Powertown.Make sure students understand that the principles above also apply to linear and exponential decay as well.

3 8-AStraightown: initially 10,000 people and growing at a rate of 500 people per yearYearStraightown10,000110,5002310152040

4 8-AStraightown: initially 10,000 people and growing at a rate of 500 people per yearYearStraightown10,000110,500211,000310152040

5 8-AStraightown: initially 10,000 people and growing at a rate of 500 people per yearYearStraightown10,000110,500211,000311,50010152040

6 8-AStraightown: initially 10,000 people and growing at a rate of 500 people per yearYearStraightown10,000110,500211,000311,50010(10x500) =15000152040

7 8-AStraightown: initially 10,000 people and growing at a rate of 500 people per yearYearStraightown10,000110,500211,000311,50010(10x500) =1500015(15x500) =175002040

8 8-AStraightown: initially 10,000 people and growing at a rate of 500 people per yearYearStraightown10,000110,500211,000311,50010(10x500) =1500015(15x500) =1750020(20x500) =2000040

9 8-AStraightown: initially 10,000 people and growing at a rate of 500 people per yearYearStraightown10,000110,500211,000311,50010(10x500) =1500015(15x500) =1750020(20x500) =2000040(40x500) =30000

10 8-APowertown: initially 10,000 people and growing at a rate of 5% per yearYearPowertown10,000110000 x (1.05) = 10,5002310152040

19 Two Basic Growth PatternsLinear Growth (Decay) occurs when a quantity increases (decreases) by the same absolute amount in each unit of time.Example: Straightown each yearExponential Growth (Decay) occurs when a quantity increases (decreases) by the same relative amount—that is, by the same percentage—in each unit of time.Example: Powertown: -- 5% each yearMake sure students understand that the principles above also apply to linear and exponential decay as well.

20 Linear/Exponential Growth/Decay?The number of students at Wilson High School has increased by 50 in each of the past four years.Which kind of growth is this?Linear GrowthIf the student populations was 750 four years ago, what is it today?4 years ago: 750Now (4 years later): (4 x 50) = 950Make sure students understand that the principles above also apply to linear and exponential decay as well.

21 Linear/Exponential Growth/Decay?The price of milk has been rising with inflation at 3.5% per year.Which kind of growth is this?Exponential GrowthIf the price was $1.80/gallon two years ago, what is it now?2 years ago: $1.80/gallonNow (2 years later): $1.80 × (1.035)2= $1.93/gallon

22 Linear/Exponential Growth/Decay?Tax law allows you to depreciate the value of your equipment by $200 per year.Which kind of growth is this?Linear DecayIf you purchased the equipment three years ago for $1000, what is its depreciated value now?3 years ago: $1000Now (3 years later): $1000 – (3 x 200)= $400

23 Linear/Exponential Growth/Decay?The memory capacity of state-of-the-art computer hard drives is doubling approximately every two years.Which kind of growth is this?[doubling means increasing by 100%]Exponential GrowthIf the company’s top of the line drive holds 300 gigabytes today, what will it hold in six years?Now: 300 gigabytes2 years: 600 gigabytes4 years: 1200 gigabytes6 years: 2400 gigabytes

24 Linear/Exponential Growth/Decay?The price of DVD recorders has been falling by about 25% per year.Which kind of growth is this?Exponential DecayIf the price is $200 today, what can you expect it to be in 2 years?Now: $2002 years: 200 x (0.75)2= $112.50

25 8-AMore PracticeThe population of Danbury is increasing by 505 people per year. If the population is 15,000 today, what will it be in three years?16,515During the worst periods of hyper inflation in Brazil, the price of food increased at a rate of 30% per month. If your food bill was $100 one month during this period, what was it two months later?$169The price of computer memory is decreasing at a rate of 12% per year. If a memory chip costs $80 today, what will it cost in 2 years?$61.95

26 8-AThe Impact of DoublingParable 1 From Hero to Headless in 64 Easy StepsParable 2 The Magic PennyParable 3 Bacteria in a Bottle

27 From Hero to Headless in 64 Easy StepsParable 1From Hero to Headless in 64 Easy StepsParable 1 “If you please, king, put one grain of wheat on the first square of my chessboard,” said the inventor. “ Then place two grains on the second square, four grains on the third square, eight grains on the fourth square and so on.” The king gladly agreed, thinking the man a fool for asking for a few grains of wheat when he could have had gold or jewels.

31 From Hero to Headless in 64 Easy StepsParable 1From Hero to Headless in 64 Easy StepsParable 1 “If you please, king, put one grain of wheat on the first square of my chessboard,” said the inventor. “ Then place two grains on the second square, four grains on the third square, eight grains on the fourth square and so on.” The king gladly agreed, thinking the man a fool for asking for a few grains of wheat when he could have had gold or jewels.264 – 1 = 1.8×1019 =≈ 18 billion, billion grains of wheatThis is more than all the grains of wheat harvested in human history.The king never finished paying the inventor and according to legend, instead had him beheaded.

32 Parable 2 The Magic PennyParable 2 A leprechaun promises you fantastic wealth and hands you a penny. You place the penny under your pillow and the next morning, to your surprise, you find two pennies. The following morning 4 pennies and the next morning 8 pennies. Each magic penny turns into two magic pennies.

33 8-AParable 2DayAmount under pillow$0.011$0.022$0.043$0.084$0.16. . .Although many financial institutions will provide the individual with an amortization schedule upon initiating the loan, there may be two important reasons that students should be able to follow the simple mathematics of a schedule. 1) Once they gain a little confidence with the flow of the columns, they are in a position to modify their own schedules using Excel or some other software and customizing it to facilitate their personal strategies of making additional payments of principal. 2) It would also be important to periodically check the accuracy of the lending institution’s monthly or annual statements to verify that everything is legitimate and up to date.

34 8-AParable 2DayAmount under pillow$0.01$0.01 = $0.01×201$0.02$0.02 = $0.01×212$0.04$0.04 = $0.01×223$0.08$0.08 = $0.01×234$0.16$0.16 = $0.01×24. . .t$0.01×2tAlthough many financial institutions will provide the individual with an amortization schedule upon initiating the loan, there may be two important reasons that students should be able to follow the simple mathematics of a schedule. 1) Once they gain a little confidence with the flow of the columns, they are in a position to modify their own schedules using Excel or some other software and customizing it to facilitate their personal strategies of making additional payments of principal. 2) It would also be important to periodically check the accuracy of the lending institution’s monthly or annual statements to verify that everything is legitimate and up to date.

35 Parable 2 1 week (7 days) $0.01×27= $1.28 2 weeks (14 days)TimeAmount under pillow1 week (7 days)$0.01×27= $1.282 weeks (14 days)1 month (30 days)50 daysAlthough many financial institutions will provide the individual with an amortization schedule upon initiating the loan, there may be two important reasons that students should be able to follow the simple mathematics of a schedule. 1) Once they gain a little confidence with the flow of the columns, they are in a position to modify their own schedules using Excel or some other software and customizing it to facilitate their personal strategies of making additional payments of principal. 2) It would also be important to periodically check the accuracy of the lending institution’s monthly or annual statements to verify that everything is legitimate and up to date.

36 Parable 2 1 week (7 days) $0.01×27= $1.28 2 weeks (14 days)TimeAmount under pillow1 week (7 days)$0.01×27= $1.282 weeks (14 days)$0.01×214= $163.841 month (30 days)50 daysAlthough many financial institutions will provide the individual with an amortization schedule upon initiating the loan, there may be two important reasons that students should be able to follow the simple mathematics of a schedule. 1) Once they gain a little confidence with the flow of the columns, they are in a position to modify their own schedules using Excel or some other software and customizing it to facilitate their personal strategies of making additional payments of principal. 2) It would also be important to periodically check the accuracy of the lending institution’s monthly or annual statements to verify that everything is legitimate and up to date.

37 Parable 2 1 week (7 days) $0.01×27= $1.28 2 weeks (14 days)TimeAmount under pillow1 week (7 days)$0.01×27= $1.282 weeks (14 days)$0.01×214= $163.841 month (30 days)$0.01×230= $10,737,418.2450 daysAlthough many financial institutions will provide the individual with an amortization schedule upon initiating the loan, there may be two important reasons that students should be able to follow the simple mathematics of a schedule. 1) Once they gain a little confidence with the flow of the columns, they are in a position to modify their own schedules using Excel or some other software and customizing it to facilitate their personal strategies of making additional payments of principal. 2) It would also be important to periodically check the accuracy of the lending institution’s monthly or annual statements to verify that everything is legitimate and up to date.

38 Parable 2 1 week (7 days) $0.01×27= $1.28 2 weeks (14 days)TimeAmount under pillow1 week (7 days)$0.01×27= $1.282 weeks (14 days)$0.01×214= $163.841 month (30 days)$0.01×230= $10,737,418.2450 days$0.01×250= $11.3 trillionAlthough many financial institutions will provide the individual with an amortization schedule upon initiating the loan, there may be two important reasons that students should be able to follow the simple mathematics of a schedule. 1) Once they gain a little confidence with the flow of the columns, they are in a position to modify their own schedules using Excel or some other software and customizing it to facilitate their personal strategies of making additional payments of principal. 2) It would also be important to periodically check the accuracy of the lending institution’s monthly or annual statements to verify that everything is legitimate and up to date.

39 Parable 2 The Magic PennyParable 2 A leprechaun promises you fantastic wealth and hands you a penny. You place the penny under your pillow and the next morning, to your surprise, you find two pennies. The following morning 4 pennies and the next morning 8 pennies. Each magic penny turns into two magic pennies. WOW!The US government needs to look for a leprechaun with a magic penny.

40 Parable 3 Bacteria in a BottleParable 3 Suppose you place a single bacterium in a bottle at 11:00 am. It grows and at 11:01 divides into two bacteria. These two bacteria each grow and at 11:02 divide into four bacteria, which grow and at 11:03 divide into eight bacteria, and so on.Question0: If the bottle is full at NOON, how many bacteria are in the bottle?Question1: When was the bottle half full?Question2: If you (a mathematically sophisticated bacterium) warn of impending disaster at 11:56, will anyone believe you?Question3: At 11:59, your fellow bacteria find 3 more bottles to fill. How much time have they gained for the bacteria civilization?

41 Single bacteria in a bottle at 11:00 am 2 bacteria at 11:01 Question0: If the bottle is full at NOON, how many bacteria are in the bottle?Single bacteria in a bottle at 11:00 am2 bacteria at 11:014 bacteria at 11:028 bacteria at 11:03. . .At 12:00 (60 minutes later) the bottle is full and contains ≈ 1.15 x1018Make sure students understand that the principles above also apply to linear and exponential decay as well.

42 Question1: When was the bottle half full?Single bacteria in a bottle at 11:00 am2 bacteria at 11:014 bacteria at 11:028 bacteria at 11:03. . .Bottle is full at 12:00 (60 minutes later)and so is 1/2 full at 11:59 amMake sure students understand that the principles above also apply to linear and exponential decay as well.

43 8-AQuestion2: If you (a mathematically sophisticated bacterium) warn of impending disaster at 11:56, will anyone believe you?½ full at 11:59¼ full at 11:58⅛ full at 11:57full at 11:56At 11:56 the amount of unused space is 15 times the amount of used space.Make sure students understand that the principles above also apply to linear and exponential decay as well.Your mathematically unsophisticated bacteria friends will not believe you when you warn of impending disaster at 11:56.

44 enough bacteria to fill 1 bottle at 12:00 Question3: At 11:59, your fellow bacteria find 3 more bottles to fill. How much time have they gained for the bacteria civilization?There are . . .enough bacteria to fill 1 bottle at 12:00enough bacteria to fill 2 bottles at 12:01enough bacteria to fill 4 bottles at 12:02Make sure students understand that the principles above also apply to linear and exponential decay as well.They have gained only 2 additional minutes for the bacteria civilization.

45 Question4: Is this scary?By 1:00- there are 2120 bacteria.This is enough bacteria to cover the entire surface of the Earth in a layer more than 2 meters deep!After 5 ½ hours, at this rate . . .the volume of bacteria would exceed the volume of the known universe.Make sure students understand that the principles above also apply to linear and exponential decay as well.Yes, this is scary!

46 Key Facts about Exponential Growth• Exponential growth cannot continue indefinitely. After only a relatively small number of doublings, exponentially growing quantities reach impossible proportions.• Exponential growth leads to repeated doublings. With each doubling, the amount of increase is approximately equal to the sum of all preceding doublings.Discussing each of the parables from the text are vital. Students should not miss this content and all the ramifications to the quantitative world around us.